Can an AI be made to maintain a train of thought?
First, for almost every question of the form "Can AI be made to X", the most obvious and straightforward answer is something like "We don't know. Probably, but if it hasn't been done yet, we're really not sure."

It's also important to understand that, from a technology standpoint, there isn't one "thing" called "AI".  There are many, many different technologies, which are loosely related (at best) and are generally lumped together under the overall rubric of "Artificial Intelligence".

All of that said, yes, there has been work on adding memory, even long-term memory, to various kinds of "AI".  The most notable recent example is the advent of LSTM in recurrent neural networks.  

Additionally, some of the work done on "cognitive architectures" has focused on the use of memory.  For more info on that, look up ACT-R and/or SOAR and read some of those papers.  

What isn't clear to me offhand, is whether or not anybody has tried applying any of these techniques to chat-bots in particular.  I wouldn't be surprised if somebody had, but I can't cite any such research off the top of my head.